Spread Codes and Spread Decoding in Network Coding

1 Spread Codes and Spread Decoding in Netw ork Coding Felice Manganiello, Elisa Gorla and Joachim Rosenthal Mathematics Ins titute Univ ersity of Zu rich W interthurerstr 19 0 CH-8057 Zurich, Switzerland www.math.u zh.ch/aa Abstract — In t his paper we introduce the class of Spread Codes for the use in ra ndom network co ding. Spread Codes are based on the constructio n of spreads in finite projective geometry . The major contribution of the paper is an ef ficient decoding algorit hm of spread codes up to half the minimum distance. I . I N T RO D U C T I O N In [KK07] K ¨ otter and Ks chischa ng develop a novel framew ork for random network coding. In this frame- work information is encoded in subs paces of a giv en ambient space over a finite field. A n atural me tric is introduced where two s ubspac es are ‘close to eac h other’ as soon as their dimension of intersection is large. This n ew frame work pos es new ch allenges to des ign new c odes with large d istances and to come up with efficient decoding algorithms. Several new papers have been written on the topic and we mention [SKK07] and [MU07]. In this paper we stud y the c lass of s preads from finite projectiv e geo metry (see e. g. [Hir98]) for pos sible u se in n etwork coding the ory . A spread S is a partition of a vector spac e by subspa ces of a fixed dimension. Elements of a sp read are subs paces of a fixed vec tor space F n q which pa irwise only intersect in the o rigin. The c odewords d eri ved in this way a re all sub space s of the sa me dimension. In other words the s pread S is a subset of the finite Grassmannia n G( k , F n q ) consis ting of all k -dimens ional subspa ces in F n q . W e will call the obtaine d co de a Spread code . Since two diff erent elements o f S only intersect in the o rigin the sprea d code S has maximal poss ible distanc e a mong a ll subsets of G( k , F n q ) . First and t hird author were partially supported by S wiss National Science Foundation under Grant no. 113251. S econd Author was supported by the Forschung skredit of the Uni versity of Zurich unde r Grant no. 5710 4101 and by the Swiss National Sci ence Foundation under Grant no. 107887. The p aper is structured as follo ws. In the n ext section we will explain the construction of spreads and we deri ve some basic properties. In Section 3 the main results of the paper are giv en. W e p rovide a n ef ficient deco ding algorithm for spread c odes esse ntially ‘up to half the minimum d istance’ with its c omplexity . The deco ding algorithm requ ires me thods from linear algebra and the application o f the Euclidean a lgorithm. I I . A L G E B R A I C C O N S T R U C T I O N O F A S P R E A D C O D E Let F q be the fin ite field with q elements. W e denote with G( k , F n q ) the Grassmann ian of a ll k -dimens ional subspa ces of F n q . Following [KK07] we define a distance function d : G( k , F n q ) × G( k , F n q ) → Z + through: d ( A, B ) := dim( A + B ) − dim( A ∩ B ) (1) = dim( A ) + dim( B ) − 2 dim( A ∩ B ) . It ha s been obs erved in [KK07] that d ( A, B ) satisfies the axioms of a metric on the finite Grassmannian G( k , F n q ) . A c onstant-dimension code S ⊂ G( k, F n q ) ha s maximal possible minimum distan ce a s long as the intersection of two dif ferent c odewords of S is tri v ial. If two subspa ces A, B ⊂ F n q intersect on ly in the zero vector then the correspond ing s ubspa ces of projective sp ace are non- intersecting. Bas ed on this we will call A, B ⊂ F n q nonintersecting s ubspac es as long a s they intersect only in the zero vector . W e want to construct a n MDS-like code S ⊂ G( k , F n q ) , i.e. c ode having maximum po ssible distan ce and maximum number of elements. In order to do this we need to restrict ou r k , n ∈ N to some pa rticular cases. It is a well known resu lt that there exists an S ⊂ G( k, F n q ) that partitions F n q (i.e. there is no vector in F n q which does not lie in a sub space ) a nd such that any tw o elements of S are non intersecting if an d only if k divides n . T hose subsets are called s preads an d this resu lt can be found in [Hir98]. 2 Consider the cas e n = r k . Let also p ∈ F n q [ x ] be an irreducible polynomial of degree k . If we deno te with P the k × k co mpanion matrix of p over F q , it follo ws that the F q -algebra F q [ P ] ⊂ Mat k × k ( F q ) is isomorphic to the finite field F q k . Deno ting with 0 k , I k ∈ Mat k × k ( F q ) respecti vely the zero and the identity matrix and given the a bove assump tions, we are ready to s tate the follo wing the orem. Theorem 1: The collection o f su bspac es S := r [ i =1 { ro wsp [0 k · · · 0 k I k A i +1 · · · A r ] | A i +1 , . . . , A r ∈ F q [ P ] } ⊂ G( k, F n q ) is a sprea d of F n q . Pr oof: T he cardinality of S is exactly the maximum number of k -dimensional n onintersecting subspac es of F n q , i.e. q n − 1 q k − 1 = q k ( r − 1) + q k ( r − 2) + · · · + q + 1 . It remains to be shown that any pa ir of subspac es in S do only intersec t tri vially that is equiv alent to showing that the 2 k × n matrix obtained p utting toge ther two matrices generating two different subspace s is full-rank. W e h av e only two cas es. T he first where the matrices I k are no t placed at the s ame column “lev el”. In this case we c an find a full-rank su bmatrix o f the form  I k A 0 k I k  . The secon d ca se is when matrices I k are at the same “lev el”. Th ere exists a sub matrix of the form  I k A 1 I k A 2  where A 1 , A 2 ∈ F q [ P ] and A 1 6 = A 2 . It follows that the determinant of the above matrix is equ al to det( A 1 − A 2 ) and is no nzero since A 1 6 = A 2 . Is it poss ible to find a previous and less gene ral version of this theo rem in [CGR07]. Definition 2: L et p be a n irreducible polynomial of degree k over F q . A spr e ad code S is a subse t o f G( k , F n q ) cons tructed as in the previous theorem. Fol- lowi ng the d efinition of [KK07] a s pread code is a q -ary code of type [ n, k , log q  q n − 1 q k − 1  , 2 k ] . Remark 3 : Spread co des are re lated to the Ree d- Solomon-like code s over Grasmannians p resented in the paper [KK07]. Following the nota tion of [KK07], let l = k and m = n − k . From the construction of Theorem 1, if follo ws that the subset o f S with i = 1 is a subcode of Reed-Solomo n-like code s. Moreover , ou r costruction provides more codewords arising from the c ases where i > 1 . There is an algebraic g eometric way to view the spreads we just introduced. For this identify the se t of polynomials in F q [ x ] having degree at mos t k − 1 with the field F q k . Co nsider the n atural isomorphism ϕ : F q k → F q [ P ] f 7→ f ( P ) . This isomo rphism induce s the natural e mbedding ˜ ϕ : G( l, F m q k ) → G( kl , F k m q ) with ˜ ϕ    ro wsp    f 11 . . . f 1 m . . . . . . f l 1 . . . f lm       = rowsp    f 11 ( P ) . . . f 1 m ( P ) . . . . . . f l 1 ( P ) . . . f lm ( P )    . The follo wing theorem is then not dif fi cult t o establish. Theorem 4: If S ⊂ G( l, F m q k ) is a spread of F m q k then ˜ ϕ ( S ) ⊂ G( kl , F k m q ) is a sp read of F k m q . Clearly G(1 , F r q k ) is a spread itself and it therefore follo ws that the subse t defined in Theorem 1 is a sp read of F n q as we ll. I I I . D E C O D I N G A L G O R I T H M W e w ill continue restricting ou r study to the case where n = 2 k and k is odd. F rom now on we will consider fixed the irreducible po lynomial p ∈ F q [ x ] . In a first step we want to es tablish a simple alge- braic criterion which cha racterizes the sprea d code S ⊂ G( k , F 2 k q ) . For this a ssume that C 1 , C 2 ∈ Mat k × k ( F q ) are matrices su ch that C := ro wsp[ C 1 C 2 ] ∈ G( k , F 2 k q ) . If C 1 is not in vertible then C ∈ S if and on ly if C 1 = 0 k . If C 1 is in vertible then C ∈ S if and only if A := ( C 1 ) − 1 C 2 ∈ F q [ P ] . W e the refore e stablish a c riterion which guarantees that a matrix A is in F q [ P ] . Let F q k be the splitting field of p over F q and S ∈ Gl k ( F q k ) be an in vertible matrix diagonalizing the matrix P , i.e. D := S P S − 1 =      λ λ q . . . λ q k − 1      where λ ∈ F q k is a root of p . Lemma 5 : Let A ∈ Mat k × k ( F q ) . The n A ∈ F q [ P ] if and o nly if AP = P A . 3 Pr oof: If A ∈ F q [ P ] then clearly AP = P A . As- sume now AP = P A and S P S − 1 = D . Since the e igen- values of P are p airwise different and D ( S AS − 1 ) = ( S AS − 1 ) D it follows tha t S AS − 1 is a diagonal matrix as well with diagon al entri es in F q k . Let { 1 , γ , . . . , γ k − 1 } be a ba sis of F q k over F q . On e ha s an expansion : S AS − 1 = k − 1 X i =0 c i D i = k − 1 X i =0 k − 1 X j =0 c i,j γ j D i with c i ∈ F q k and c i,j ∈ F q . Equiv alently we have: A = k − 1 X j =0 k − 1 X i =0 c i,j P i ! γ j . It follows that A = P k − 1 i =0 c i, 0 P i and A ∈ F q [ P ] . The follo wing gi ves an algebraic criterion for checking when a sub space is a c odeword. Cor ollary 6: The subspac e rowsp[ I k A ] ∈ G( k , F 2 k q ) is a c odeword of S if and o nly if S AS − 1 is a diagon al matrix. W e state now the unique deco ding problem. As - sume C := ro wsp[ C 1 C 2 ] ∈ S was sent and R := ro wsp[ R 1 R 2 ] ∈ G( k, F 2 k q ) was received. If dim( C ∩ R ) ≥ k + 1 2 (2) then unique de coding is poss ible. In the s equel we will consider the received sub space R ∈ G( k, F 2 k q ) such that there exists a codeword C ∈ S such that (2) holds . A. C ase R 1 not invertible. Let R an d C be su bspace s satisfying the con dition (2). The goa l of this subse ction is to ana lyze the be havior of the decoding problem whe n R 1 is n ot in vertible. This situation splits in two dif ferent ones. The first one is wh en 0 ≤ rank( R 1 ) ≤ k − 1 2 . The closes t codeword in this c ase is only the subsp ace ro w s p[0 k I k ] . The second case is characterized by k +1 2 ≤ rank( R 1 ) ≤ k − 1 . W ith the follo wing lemma we bring back the d ecoding problem of the s ubspac e R to the one of a subsp ace ˜ R c lose relate d to R a nd lying in the same ball with cen ter in the codeword C . Lemma 7 : Let R ∈ G( k , F 2 k q ) suc h that k +1 2 ≤ rank( R 1 ) ≤ k − 1 and C ∈ S suc h that (2) holds. Then there exists a subspac e ˜ R := rowsp[ ˜ R 1 ˜ R 2 ] ∈ G( k, F 2 k q ) satisfying: • ˜ R 1 is in vertible, • dim( R ∩ ˜ R ) = rank( R 1 ) , a nd • dim( C ∩ ˜ R ) ≥ k +1 2 . Pr oof: Let t := rank( R 1 ) . Row redu cing the matrix [ R 1 R 2 ] we obtain the matrix  ¯ R 1 ¯ R 2 0 E  where ¯ R 1 , ¯ R 2 ∈ Mat t × k ( F q ) with R 1 fullrank a nd 0 , E ∈ Mat k − t × k ( F q ) where 0 is the zero matrix. Since ro wsp[0 E ] ⊂ ro wsp[0 k I k ] we deduce tha t dim( C ∩ ro wsp[0 E ]) = 0 . It follows immediately that dim( C ∩ ro wsp[ ¯ R 1 ¯ R 2 ]) = dim( C ∩ ˜ R ) ≥ k + 1 2 . The matrix represen ting the s ubspac e ˜ R can then be constructed a s follows: • ˜ R 1 is the completion o f the matrix ¯ R 1 to a n in vertible matrix, a nd • ˜ R 2 is the completion of the ¯ R 2 to a k -s quare matrix by ad ding rows of zeros. Cor ollary 8: The s olution to the u nique decod ing problem for both subspa ces R an d ˜ R consists of the same cod ew ord C ∈ S . B. C ase R 1 in ve rtible. W e c an now construct an algorithm for the uniqu e decoding p roblem of subs paces with R 1 in vertible. Theorem 9: Le t R := ro wsp[ R 1 R 2 ] ∈ G( k, F 2 k q ) a subspa ce with R 1 in vertible. Then there exists a unique matrix A ∈ F q [ P ] and a unique matrix N ∈ Mat k × k ( F q ) of ran k at most k − 1 2 such that R − 1 1 R 2 = A + N . In this cas e ro wsp[ I k A ] is the closest codeword to R in the distanc e (1). Pr oof: The unique ness follows from the d istance properties of the co de. As sume r o wsp[ I k A ] be the closest cod ew ord to R . Since ro wsp  I k A R 1 R 2  = rowsp  I k A 0 k R − 1 1 R 2 − A  has dimension at most 2 k − k +1 2 = k + k − 1 2 it follo ws that the matrix N := R − 1 1 R 2 − A has rank a t most k − 1 2 . Cor ollary 10: Let R := rowsp[ R 1 R 2 ] ∈ G( k , F 2 k q ) a subspa ce with R 1 in vertible. Let Y := S ( R − 1 1 R 2 ) S − 1 . Then there is a un ique polynomial f ∈ F q [ x ] with deg f < k such that Y − f ( D ) ha s rank at most k − 1 2 . Pr oof: The existence follows directly from the las t theorem. Conc erning the uniquenes s a ssume that Y = f 1 ( D ) + N 1 = f 2 ( D ) + N 2 . It the n follows that R − 1 1 R 2 = f 1 ( P ) + S − 1 N 1 S = f 2 ( P ) + S − 1 N 2 S and becau se of the uniquenes s part of Th eorem 9 the result follows. 4 The algorithm extrapolates the ev a luations of the poly- nomial f ∈ F q [ x ] from the matrix Y − f ( D ) . O nce the polynomial f ∈ F q [ x ] is found, its ev aluation at P gi ves us the matrix A ∈ F q [ P ] such tha t ro wsp [ I k A ] is the codeword close st to R . Notice tha t the co efficients of f are exactly the co efficients of the expre ssion of f ( λ ) in the basis { 1 , λ, . . . , λ k − 1 } of F q k over F q . The following two rema rks from finite fie ld the ory (see [LN94]) will be important. First, given any f ∈ F q [ x ] and any µ ∈ F q k , then f ( µ q ) = f ( µ ) q . Secon d, giv en a finite field F q with q elements it holds x q − x = Y α ∈ F q ( x − α ) . W e outline n ow the comp lete dec oding algorithm. Let R := ro wsp [ R 1 R 2 ] be the rece i ved s ubspa ce satisfying cond ition (2). Ass ume that R 1 is in vertible. Compute Y := S ( R − 1 1 R 2 ) S − 1 . If the ma trix Y i s diagonal, then R is already a codew ord of S by Corollary 6. Otherwise the matrix Y − f ( D ) is o f the form      y 1 , 1 − f ( λ ) y 1 , 2 · · · y 1 ,k y 2 , 1 y 2 , 2 − f ( λ q ) · · · y 2 ,k . . . . . . . . . . . . y k , 1 y k , 2 · · · y k ,k − f ( λ q k − 1 )      =      y 1 , 1 − f ( λ ) y 1 , 2 · · · y 1 ,k y 2 , 1 y 2 , 2 − f ( λ ) q · · · y 2 ,k . . . . . . . . . . . . y k , 1 y k , 2 · · · y k ,k − f ( λ ) q k − 1      where s ome entries off of the diag onal are nonzero. Denote by X the matrix obtained from Y − f ( D ) by substituting x for f ( λ ) . By Co rollary 10 there exists a unique value for x ∈ F q k (namely x = f ( λ ) ) s uch that rank( X ) ≤ k − 1 2 . T he decod ing problem redu ces to finding s uch a value. The c ondition on the rank is equi valent to having all minors of size k +1 2 of the matrix X being ze ro. This giv es us a system of univ a riate equa tions which apriori may be hard to solve. Howe ver since the sys tem ha s a unique solution, every minor is divisible by ( x − f ( λ )) . Hence in order to fin d f ( λ ) it su f fices to compute the gcd of the field equation x q k − x with enough e quations from our sy stem. More prec isely we look for a nonze ro minor of size k − 1 2 which do es not in volv e a ny diagon al entry . If no su ch minor exists, then look for a nonze ro minor of smaller size wh ich aga in does not in volve any diagonal entry . Let t be the size of the minor . Comple te the c orrespond ing size t s ubmatrix to a s ubmatrix of X of size k +1 2 . No tice tha t this can be done by adding k +1 2 − t rows and columns with the same index. Th e determinant of this submatrix is a n onzero polyno mial m ∈ F q k [ x ] which has f ( λ ) as a root. Apply the Euc lidean Algorithm in order to comp ute g := gcd( x q k − x, m ) . If the degree o f g is sma ll, c ompute its roots and substitute them in X in order to find f ( λ ) . Otherwise compute ano ther minor in the same way as for the previous one . Procee d by computing the gcd of this polynomial with g . The algorithm ends once it find s f ( λ ) . C. Co mplexity The overall complexity of the algorithm is dominated by the Euclidea n Algorithm. In the worst ca se scenario, i.e. wh en the maximal nonze ro minor o f f d iagonal ha s size 1, the algorithm’ s complexity is O ( q k log 2 3 log q k ) in F q k . The complexity could be dras tically dec reased by the following c onjecture: for every error matrix N ∈ Mat k × k ( F q ) of rank t ≤ k − 1 2 there exists a non zero minor of size t of the ma trix X which do es not in volv e any diagonal entry . Consider now such a non zero minor of X and extend the related su bmatrix adding one row and one c olumn with the same index. The determinant of this subma trix leads to an equation of the typ e x q i = α with α ∈ F q . Raising bo th s ides o f the eq uation to the q k − i -th power and using the field equation of F q k we get: x = α q k − i . Using the Repeated Squa ring Algorithm for computing powers in F q k , the co mplexity of the d ecoding a lgorithm decreas es to O (log q k − i ) = O ( k − i ) ope rations in F q k . A reference for efficient algorithms is [GG03]. In particular s ee Sec tion 4.3 for the Re peated S quaring Algorithm, Section 11.1 for performing the Euclidea n Algorithm, Chapter 14 for factoring univ ariate po lyno- mials a nd Section 25.5 for computing de terminants. D. No n-perfectne ss of a Spread Code Spreads are perfect in the sens e that every non zero vector of F n q is in one and only one sub space of the spread. In c oding theory a code is perfec t if the total amb ient space is covered with the balls centere d in the cod ew ords and having radius h alf the minimum d istance. It arises the qu estion if sprea d code s are perfect in this sense . The answe r turns out to be nega ti ve in gene ral and this result c an be found in [MZ95]. 5 A C K N O W L E D G M E N T S W e would like to thank Joa n Jos ep Climent, Fe lix Fontein, V er ´ onica Requena and Jens Zumbr ¨ ag el for many helpful d iscussion s du ring the preparation of this p aper . R E F E R E N C E S [CGR07] J. J. Cl iment, F . J. Garcia, and V . Requena. On the construction of bent functions of 2k variables from a primitiv e polyno mial of degree k. preprint, 2007. [GG03] J. von zur Gathen and J. Gerhard. Modern computer algebr a . Cambridge Uni versity Press, Cambridge, second edition, 2003. [Hir98] J. W . P . Hi rschfeld. Pro jective Geometries over Finite F ields . Oxford M athematical Mono graphs. The Clarendon Press Oxford Univ ersity Press, New Y ork, second edition, 1998. [KK07] R. K oetter and F . Kschischang. Coding for errors and erasures in random network coding. submitted, 2007. [LN94] R. Lidl and H. Niederreiter . Introd uction to F inite Fields and their A pplications . Cambridge University Press, Cam - bridge, London, 1994. Re vised edition. [MU07] A. Montanari and R. Urbanke. Coding for network coding. submitted, 2007. [MZ95] W . J. Martin and X. J. Zhu. Anticodes for the grassman and bilinear forms graphs. Designs, Codes and Cryptogr aphy , 6(1):73–79, July 1995. [SKK07] D. S ilva , F . Kschischang, and F . K ¨ otter . A rank-metric approach to error control in random network coding. submitted, 2007.